Mathematical modeling of cardiac cells and tissues at different levels of abstraction.
- Emergent properties of myofilaments
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Emergent properties of myofilaments
Myofilaments comprise the majority of both skeletal and cardiac muscles and are the primary components involved in active contraction. Skeletal muscle provides the means for motion in our bodies and places the largest demands on our cardiovascular system. Equally important, cardiac muscle is essential for life as it pumps blood to supply all parts of the body with oxygen and essential nutrients. While cells comprise the basic, unitary structure of the heart, the sarcomere represents the basic, repeating subcellular unit of the myofilament. The sarcomere is composed primarily of interdigitated thick and thin filaments that interact and generate force when activated. The thin filament is a long polymer composed of a double helix of actin monomers. The thick filament is composed of myosin, a molecular motor that converts the energy from adenosine triphosphate (ATP) hydrolysis into a net motion of the myosin head region with respect to the actin monomers in the thin filament. The movement of the head region is assumed to stretch an extensible neck region that connects the head region to the thick filament, an event that produces a net force between the thick and thin filaments. The attached myosin heads with the extensible neck regions form links between thick and thin filaments. These links were detected via electron micrographs in the early 1950's by Huxley who termed them "crossbridges" although details such as head and neck regions could not be resolved in these inital studies. For a good online overview, see these lecture notes from Pieter de Tombe.
Figure 1: Schematic representation of the repeating sarcomere structure in striated muscle is shown in (a). The sarcomere is defined from z-line to z-line with interdigitated thick and thin filaments that can interact to produce force. Activated muscle generated force in the direction to pull thin filament to the center of the sarcomere. The inset shows half of a thick filament, composed of myosin, that interacts with a single thin filament, composed primarily of two-stranded helix of actin. Myosin heads can bind to specific sites on actin (see crosshatched example) and then rotate to stretch the extensible neck region (shown as a stretched link in conjunction with the solid head in the diagram). Rotated heads generate a net force between thick and thin filaments. Regulatory units bind calcium (Ca) which in turn biases the units to take a permissive conformation (shown schematically as a raised location) to allow myosin to bind. The regulatory units physically interact so neighboring units will tend to align with neighbors to have similar conformations (for details, see Hussan et al., 2006). Illustrations of the component proteins of the thick and thin filaments are shown in (b). Details of thin filament proteins show a two strand helix of actin monomers with regulatory units (troponin and tropomyosin). Tropomyosin is indicated by the thin light-green line. Troponin is composed of three subunits, TnC, TnI and Tn. The thick filament is composed of the intertwined tails of myosin from which the head and neck region extend to interact with the thin filament. The figure reprinted from Hussan et al., 2006.Special thanks to Lei Jin who contributed artwork of sarcomere proteins.
While muscle physiology has been studied for over 100 years, large gaps still remain in our understanding of the fundamental processes of muscle contraction. Specifically, we have an incomplete understanding of how individual molecular-level myosin motors produce force from the hydrolysis ATP. Recent characterization using X-ray crystallography have been incorporated into an animation from Ron Milligan's Lab that illustrates the cyclical interactions of actin and myosin. However, note that this is just an animation based on static data, and we do not yet have the ability to resolve such motions on the molecular scale. Other researchers are using techniques such as laser traps, but these data are often difficult to interpret. One use of mathematical models is quantitiatively bridge data collected at different scales with different preparations, or under different experimental conditions.
Other controversies revolve around how the actin-myosin interactions are controlled by the regulatory proteins troponin and tropomyosin (for a good review, see Solaro and Rarick, Circ Res 83(5):471-80, 1998). Specifically, the regulatory proteins are often improperly portrayed as a simple "switch" that controls the "motor" proteins (the actin-myosin interactions described above). However, the "switch" and "motor" analogy fails to capture the interactive nature of the system. For example, strongly-bound myosin is thought to hold the regulatory proteins in a permissive conformation even in the absence of activator calcium (Ca). Hence the "motor" can hold the "switch" in the on position, at least for a short while. This interactiveness is also important in understanding the cooperative interactions that give cardiac muscle a high sensitivity to small changes in activator Ca concentration. This sensitivity, as characterized by a high Hill coefficient, is thought to arise from cooperative interactions between crossbridges and regulatory proteins, between neighboring crossbridges, between neighboring regulatory proteins, or some combination of the above.
Figure 2: The figure shows that developed force is a steep nonlinear function of activator calcium (Ca) concentration. The data in (a) is experimental data from rat cardiac muscle collected by Pieter de Tombe's laboratory. The steep calcium sensitivity is required to produce much larger relative change in force (e.g., 1000x) for a much smaller relative change in [Ca] (e.g., 10x) for each heartbeat. The calcium concentration change is small because of energetic limitations of cell's ability to move calcium ions. In contrast, force must change by a large amount to produce proper filling and ejection from the heart on every heartbeat. The figure reprinted from Hussan et al., 2006.
Spatio-temporal models are required to explicitly simulate the steric control of actin-myosin interactions by the spatial array of regulatory proteins. However, detailed, spatially-explicit models are computationally costly as well as difficult to constrain the model parameters given the paucity of direct experimental data at the molecular level. However, appropriate approximations may produce simpler models. For example, under steady-state conditions, the spatial array of regulatory proteins can be modeled using an Ising model borrowed from statistical mechanics (see Rice et al., 2003). The Functional Genomics and Systems Biology group is developing other modeling approaches to better understand how gross muscle properties emerge from the interactions at the molecular level. Our approach incorporates results from recent advances in experimental techniques that allow the fundamental interactions of the system to be probed. These techiniques include EM reconstructions that reveal proteins conformations, nucleotide substitutions that modify the behavior of myosin motors, and genetic mutations that alter the regulatory proteins.
The availability of increased computing power will make possible new classes of biological models that include detailed representations of proteins and protein complexes with spatial interactions. Along these lines, we are developing a model of the interaction of actin and myosin within one pair of thick and thin filaments in the cardiac sarcomere. The model includes explicit representations of actin, myosin, and regulatory proteins. The preliminary version of the work is detailed in Hussan et al., 2006. Although this is not an atomic-scale model, as would be the case for molecular dynamics simulations, the model seeks to represent spatial interactions between protein complexes that are thought to produce characteristic cardiac muscle responses at larger scales. While the model simulates the microscopic scale, when model results are extrapolated to larger structures, the model recapitulates complex, non-linear behavior. For example, the model recapitulates the steep sensitivity of developed force as function of activator Ca , as seen in real muscle (see Fig. 2 for real muscle response). By bridging spatial scales, the model provides a plausible and quantitative explanation for several unexplained phenomena observed at the tissue level in cardiac muscle. Model execution entails Monte Carlo based simulations of Markov representations of calcium regulation and actin-myosin interactions. While most of the results presented so far are preliminary, we propose that this model will be suitable to serve as a basis for larger-scale simulations of multiple fibers assembled into larger sarcomere structures. For example, Fig. 3 shows a possible mapping of 32-sarcomere myofibril model onto one rack of a Blue Gene L computer.
Figure 3: A possible mapping of 32-sarcomere myofibril model onto one rack of a Blue Gene L computer. Simulation of two thick and eight thin filaments will be executed on a dual-core processor. Then 64 thick and 256 thin filaments can represent a full sarcomere at the level of a node card. The mapping is approximate because a final implementation may require some redistribution of the model among computation units at the level of processors or node cards, in order to balance computational loads with given communication constraints. (DDR indicates double-data-rate synchronous dynamic random access memory; GB, gigabytes; GF, gigaflop; TF, teraflops.) The full-sarcomere illustration is adapted with permission from R.C. Wagner, Professor Emeritus of Biological Sciences, University of Delaware, Newark, Delaware. The complete figure reprinted from Hussan et al., 2006.
We also look for ways in which new data at the molecular scale can be combined with more classical data and data collected at higher levels of organization such as myofibril, muscle cell, and whole heart. In this way, we hope to produce models that can be applied across spatial scales. This is often key in understanding cardiac phenomenon where effects span extreme spatial and time scales. For example, many cardiac drugs operate on the molecular scale with effects on ionic channels, whereas one wants to understand the effects of these drugs on arrythmias and sudden cardiac death (i.e., the effects at organism and whole heart level over a much longer time scale). For this, one need for models that can span large spatial and time scales (please see Hunter, P.J. and Borg T.K. Integration from proteins to organs: the Physiome Project. Nat Rev Mol Cell Biol. 2003 Mar;4(3):237-43).
Academic collaborators
This work is undertaken in collaboration with several research labs. The lab of Pieter de Tombe at University of Illinois Chicago has provided much of the data used to generate and refine the models. The modeling work is also done in collaboration with the Laboratory of Donald Bers at Loyola University Chicago. The lab collects data on cardiac cells to characterize their electrophysiology, calcuim handling mechanism, and cellular signaling. The laboratory also used these data to construct and validate mathematical models of cardiac cells. The lab has free downloads of their rabbit ventricular cardiac cell modeling package called LabHeart (developed by Jose Puglisi) at their website. This package has a very user-friendly interface and is a great demonstrator of cardiac models. My myofilament models are not yet incorporated into this package although these should be added in the near future.
Publications from our group
Rice, J.J., Wang, F., Bers, D.M.. and De Tombe, P.P. Approximate model of cooperative activation and crossbridge cycling in cardiac muscle using ordinary differential equations, Biophysical Journal (2008) (Pubmed).
Rice, J.J., Tu, Y., Poggesi, C. and De Tombe, P.P. Spatially-Compressed Cardiac Myofilament Models Generate Hysteresis that Is Not Found in Real Muscle, Pacific Symposium on Biocomputing 13:366-377 (2008). (pdf)
Rice, J.J. and Kohl, P. The response of cardiac muscle to stretch: The role of calcium. In Electrical Diseases of the Heart: Genetics, Mechanisms, Treatment, Prevention. Gussak, I., Antzelevitch, C., Wilde, A.A.M., Friedman, P.A., Ackerman, M.J., and Shen, W.-K. eds. Springer-Verlag (2008).
Hussan, J. and de Tombe, P.P. and Rice, J.J. A spatially detailed myofilament model as a basis for large-scale biological simulations, IBM Reseach and Development, 50(6) (Issue on Systems Biology) (2006) (pdf).
Rice, J.J. and Bers, D.M. The response of cardiac muscle to stretch: The role of calcium. In Cardiac Mechano-electric Feedback and Arrythmias: From Pippette to Patient. Kohl, Franz, and Sachs, eds. Elsevier: Philadelphia (2005).
Rice, J.J. and de Tombe, P.P. Approaches to modeling crossbridges and calcium-dependent activation in cardiac muscle. Prog Biophys Mol Biol. Jun-Jul;85(2-3):179-95 (2004). (Pubmed)
Rice, J.J., Stolovitzky, G., Tu, Y., and de Tombe, P.P. Ising model of cardiac thin filament with nearest-neighbor cooperative interactions. Biophysical Journal Feb;84(2 Pt 1):897-909. (2003). (Pubmed)
Rice, J.J. and Jafri, M.S. Modeling calcium handling in cardiac cells. Philisophical Transactions of the Royal Society London, 359: 1143-1157 (2001).
Download Models
As a service to the cardiac modeling community, we are freely distrubuting source code for the models from:
Rice, J.J., Wang, F., Bers, D.M.. and De Tombe, P.P. Approximate model of cooperative activation and crossbridge cycling in cardiac muscle using ordinary differential equations, Biophysical Journal (2008) (Pubmed).
Note: There is a typo in the equations in the paper. These values are transposed. The correct values are:
kn_p = 0.50 /ms
kp_n = 0.050 /ms
Source code for the ODE-based myofilament model is available for download in the following formats:
- XPP code (implementation from published paper for Figs. 3-8)
- C++ code for combined Chicago and myofilament model (implentation for published paper for Figs. 9-10)
- Matlab implementation of the rat model (not used in the originally published paper)
XPP (code used for published paper for Figs. 3-8)
XPP is a freely distributed package to implement and execute dynamical models. While this simulation package may not be at well known as others, the code is stable, reliable, portable, well-documented, and free. XPP is a freely available package to implement and execute dynamical systems. XPP has many advantages for investigating ODE-based models such as easy searching of parameter space and multiple integration methods. The provided code implements the model and gives instructions to recreate the figures in the paper. The standard file to code and execute a model is in XPP is an .ODE file. To run this model, the user will need to install XPP on their computer from the website. Using the parameter sets provided in the ODE file, the user can recreate Figs. 3-8. More detailed instructions to recreate the figures are included in the .ODE file. Three sample output files are provided so that the user can be sure the code is executing correctly. The sample output files are plain text files and are located in the directory named: “SampleOutput”. The following link will download a zipped directory with the XPP and C++ code - CodeDistribution_XPP_CPP.zip.
C++ code for combined Chicago and myofilament model (implentation for published paper for Figs. 9-10)
This package of code implements the Chicago model of the rabbit ventricular myocyte. This directory contains a C++ source code called ChicagoMyofilement.cpp. This is the main routine to implement the combined Chicago model and myofilament model at shown in Figs. 9-10. The integration routine is CVODE that free distributed by Lawrence Livermore National Laboratories. The code was compiled and executing using the Eclipse Integrated Development Environment (IDE). This package can be downloaded for free and installed from Eclipse.Org. One can also use other compilers and a sample make file is provided as a guide. The main routine as defined in ChicagoMyofilement.cpp contains detailed instructions to recreate the Figs. 9-10. Three sample output files are provided so that the user can be sure the code is executing correctly. The subdirectory CVODE The sample output files are plain text files and are located in the directory named: “SampleOutput”. The following link will download a zipped directory with the XPP and C++ code - CodeDistribution_XPP_CPP.zip.
Matlab implentation of the rat myofilament model (not used in original publication)
This rountine implements a rat version of the myofilament model. The code is generously provided by Jason Yang of University of Virginia. The file is plain text, but has a .M extension for Matlab. The code implements a subroutine that is suitable to interface with a model of electrophysiology and Ca handling. The following link will a text file with the Matlab code - MyofilamentRat.M.
Links to other persons and groups doing related work:
Last updated 23 Sep 2009